I recently heard someone use the phrase "extract a price". He was referring to a scenario in which an action had an associated cost. To me, it seems the correct phrase should have been "exact a price". But, given the literal meaning of "extract", it seems his usage could have been correct.

2 Answers
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I tentatively agree. If I heard someone "extract a price" I would take it to mean that someone had been unwilling to give a price. "He was coy with the price but I extracted a price from him". Whereas exact a price would be to compel payment or similar.
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WudangSep 20 '11 at 17:16

‘Extract a price’ is well established. The OED’s definition 3b of 'extract' is ‘figurative; especially to draw forth (a confession, money, etc.) against a person's will.’ An 1825 citation shows the monetary use: ‘He had extracted the last extractable halfpenny.’

'Exact a price’ is also possible. The OED gives the verb ‘exact’ as meaning ‘to demand and enforce the payment of (fees, money, taxes, tolls, penalties, etc.); to extort.’

Not so. Apart from the somewhat contrived context in Wudang's comment to Jeff's answer, the standard phrase is exact a price. Here are the usage graphs for the past century, and I have to say nearly all the (relatively few) instances of extract are simply incorrect use of the phrase.
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FumbleFingersSep 20 '11 at 21:13